On three differential equations associated with Chebyshev polynomials of degrees 3, 4 and 6

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• 作者：Li Chien Shen
• 关键词：Chebyshev polynomials ; Eisenstein series ; Jacobi theta functions ; Weierstrass elliptic function
• 刊名：Acta Mathematica Sinica, English Series
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：21-36
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
• ISSN：1439-7617
• 卷排序：33

文摘

We shall study the differential equation $${y'^2} = {T_n}\left( y \right) - \left( {1 - 2{\mu ^2}} \right)$$ where μ2 is a constant, T_n(x) are the Chebyshev polynomials with n = 3, 4, 6.The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function which can be used to construct theories of elliptic functions based on ₂F₁(1/4, 3/4; 1; z), ₂F₁(1/3, 2/3; 1; z), ₂F₁(1/6, 5/6; 1; z) and provide a unified approach to a set of identities of Ramanujan involving these hypergeometric functions.